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Wick product

  1. Aug 28, 2010 #1
    We use the antisymmetric Fock space ( "fermions"). We denote by [tex]c(h)[/tex] a creator operator.

    I need to evaluate the following quantity:

    [tex]< \Omega , \big(c(h_1)+c(h_1)^{*}\big)\big(c(h_2)+c(h_2)^{*}\big) \ldots \big(c(h_n)+c(h_n)^*\big)\Omega>[/tex]

    where [tex]\Omega[/tex] is the unit vector called vaccum, [tex] <\cdot\ ,\ \cdot>[/tex] the scalar product and [tex]h_1,\ldots,h_n[/tex] any vectors.

    I need a reference or an explanation.

    Thank you!
     
  2. jcsd
  3. Aug 28, 2010 #2

    strangerep

    User Avatar
    Science Advisor

    You didn't really say what part of this is giving you trouble.

    As a broad approach, I'd try explicit evaluation for some
    low values on n, and try to see a pattern. Then try and
    prove that the pattern holds for all n via induction.
     
  4. Aug 29, 2010 #3
    If n is odd, I understand that the quantity is 0: We can write
    the quantity as a sum of monomials in which all
    creators are to the right of all annihilators (anti-Wick ordered). A
    such monomial is a product of an odd number of factors. Clearly
    the vacuum state annihilates a such monomial. We deduce the result by
    linearity.

    If n=2k is even: if a creator is to the left of a annihilator note that we have the formula

    [tex]
    <\xi,c(e)c(f)^*\eta>=0
    [/tex]

    proof:

    [tex]
    <\xi,c(e)c(f)^*\eta>_{\mathcal{F}(H)}
    = <f,e>_H<\xi,\eta>_{\mathcal{F}(H)}-<\xi,c(f)^*c(e)\eta>_{\mathcal{F}(H)}
    = <f,e>_H<\xi,\eta>_{\mathcal{F}(H)}-<f\otimes \xi,e\otimes\eta>_{\mathcal{F}(H)}
    = 0
    [/tex]

    where in the first equality, I use [tex]c(f)^*c(e)+c(e)c(f)^* =\ <f,e>_{H} Id_{\mathcal{F}(H)}[/tex].

    Now each term of the product is a sum of monomials. By the previous calculation, if a creator is
    to the left of a annihilator then the vacumm state annihilates this monomial.
    Then, we only must consider the monomial which all creators are to the right of
    all annihilators (anti-Wick ordered). Moreover, it is clear that if
    the number of creators and the number of annihilators is
    different the vacuum state annihilates the (anti-Wick ordered) monomial.
    Consequently, the quantity is equal to
    [tex]
    <\Omega ,c(h_1)^*...c(h_k)^*c(h_{k+1})..c(h_{2k})\Omega>=<h_1,h_{2k}>...<h_k,h_{k-1}>
    [/tex]
    But this last quantity is not "symmetric" ([tex]c(h_1)^*[/tex] and [tex]c(h_2)^*[/tex] anticommute).

    Then I have a BIG problem. Where is the mistake?
     
    Last edited: Aug 29, 2010
  5. Aug 30, 2010 #4

    strangerep

    User Avatar
    Science Advisor

    I'm not sure I understand your notation, I presume that

    [tex]
    |\eta\rangle ~:=~ c^*(\eta) |\Omega\rangle ~~~~~~ ?
    [/tex]

    Hmm.... permit me to simplify (i.e., abuse) the notation...

    I'll write
    [tex]
    c^*(f) ~\to~ f^* ~~~~~~\mbox{etc,}
    [/tex]
    and I'll use ordinary parentheses to denote the inner product in H, e.g., (f,g).
    I'll also use "0" for the vacuum.

    Then

    [tex]
    \langle\xi,c(e)c(f)^*\eta\rangle ~\to~ \langle 0|\, \xi \, e \, f^* \, \eta^* \, |0\rangle
    ~=~ (f,e)\,(\eta,\xi) ~-~ (f,\xi) \, (\eta,e)
    ~\ne~ 0 ~~,
    [/tex]

    unless I've made a mistake, or misunderstood your notation.
     
  6. Aug 30, 2010 #5
    Thank you for your answer.
    However, with my notations, we have
    [tex]
    c(\eta) |\Omega\rangle ~:=~ |\eta\rangle
    [/tex]
    [tex] c(\eta) [/tex] is a creator and not a annhilator.
     
  7. Aug 30, 2010 #6

    strangerep

    User Avatar
    Science Advisor

    OK, but I still don't see how you get zero...


    [tex]
    \langle\xi,c(e)c^*(f)\eta\rangle ~=~ \langle 0|\, c^*(\xi) \, c(e) \, c^*(f) \, c(\eta)\, |0\rangle
    ~=~ \langle 0| \big( (\xi, e) - c(e)c^*(\xi) \big) \big( (f,\eta) - c(\eta) c^*(f) \big) |0\rangle
    ~=~ (\xi, e)\,(f,\eta) ~\ne~ 0 ~~,
    [/tex]
     
  8. Sep 1, 2010 #7
    Thank you very much!
     
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